An upper bound for polynomial volume growth of automorphisms of zero entropy

Let $X$ be a normal projective variety of dimension $d$ over an algebraically closed field and $f$ an automorphism of $X$. Suppose that the pullback $f^{\ast }{\mathrm{\mid }}_{{\mathsf{N}}^1(X{)}_{\mathbf{R}}}$ of $f$ on the real Néron–Severi space ${\mathsf{N}}^1(X{)}_{\mathbf{R}}$ is unipotent and denote the index of the eigenvalue $1$ by $k+1$. We establish the following upper bound for the polynomial volume growth $\mathrm{p}\mathrm{l}\mathrm{o}\mathrm{v}(f)$ of $f$: $$\mathrm{p}\mathrm{l}\mathrm{o}\mathrm{v}(f)\le (k\mathrm{/}2+1)d\mathrm{.}$$ This inequality is optimal in certain cases. Moreover, we prove that $k\le 2(d-1)$, extending a result of Dinh–Lin–Oguiso–Zhang for compact Kähler manifolds to arbitrary characteristic. By combining these two inequalities, we obtain the optimal bound $$\mathrm{p}\mathrm{l}\mathrm{o}\mathrm{v}(f)\le d^2,$$ that affirmatively answers the questions of Cantat–Paris-Romaskevich and Lin–Oguiso–Zhang.